University of Arkansas, Fayetteville
ScholarWorks@UARK
Theses and Dissertations
8-2014
Multistage Accelerated Reliability Growth Testing
Model and Data Analysis
Leiying Jiang
University of Arkansas, Fayetteville
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Recommended Citation
Jiang, Leiying, "Multistage Accelerated Reliability Growth Testing Model and Data Analysis" (2014).Theses and Dissertations. 2212. http://scholarworks.uark.edu/etd/2212
Multistage Accelerated Reliability Growth Testing Model and Data Analysis
A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Industrial Engineering
by
Leiying Jiang Nanchang University
Bachelor of Science in Industrial Engineering, 2012
August 2014 University of Arkansas
This thesis/dissertation is approved for recommendation to the Graduate Council.
____________________________ Dr. Edward Pohl
Thesis Director
____________________________ ________________________________ Dr. Richard Cassady Dr. Kelly Sullivan
Abstract
Accelerated reliability growth testing has recently received a renewed interest in reliability engineering. The concepts of accelerated testing and reliability growth individually have been used in a variety of applications, either for hardware systems or software systems. The advantage of using a combined strategy is that it could shorten the testing time while maximizing the reliability. In the literature, there are many references related to optimal test design for reliability from either a component level or a system level. In this research, we suggest an approach which conducts accelerated testing at the component level while supporting estimates of reliability at the system level. Our approach helps one decide where and at what level to conduct accelerated test during the system design and testing process. Our approach is designed to reduce testing cost while still demonstrating that system level requirements are met. We do this testing at lower levels in an accelerated environment, where costs are lower, and minimize the amount of testing at the higher integrated system level where it tends to be more expensive.
Contents
Chapter 1. Introduction ... 1
1.1. Background ... 1
1.2. Motivation ... 4
Chapter 2. Literature review ... 6
2.1. Protocol Research ... 6
2.2.1. ISO acceleration factor ... 9
2.2.2. Environmental and operational acceleration factors approach ... 10
2.2.3. ARGT through critical parts ... 11
2.2.4. Chi-Squared accelerated reliability growth testing method ... 12
2.2.6. Accelerated Testing Based on the Duane Model ... 15
2.4. Source for model modification ... 18
Chapter 3. Proposed Modeling Framework ... 22
3.1. Problem statement ... 22
3.2. Notation ... 24
3.3. Assumptions ... 26
3.5. Proposed Model... 29
3.6. Implementation of Proposed Model... 32
Chapter 4. Multiple Cases Consideration ... 35
4.1. Exponential distribution ... 35
4.2. Weibull Distribution ... 39
Chapter 5. Modeling Analysis and Sensitivity Analysis ... 44
5.1. Failure intensity analysis ... 44
5.2. Acceleration factors analysis ... 46
5.3. Effectiveness factors analysis... 48
5.4. Failure modes elimination analysis ... 49
Chapter 6. Design Induced Failure Modes Model ... 52
6.1. Modified model ... 52
Chapter 7. Parameter Estimation ... 59
7.1. Estimation of normal condition failure intensity... 59
7.2. Estimation of acceleration factors ... 60
7.3. Estimation of effectiveness factors ... 61
Chapter 8. Conclusion ... 62
8.1. Conclusions ... 62
8.2. Future work ... 63
References ... 65
Appendix ... 69
Appendix 1. Protocol Research Details... 69
List of Tables
Table 1. Protocol Research Criteria ... 7
Table 2. Decrease Failure Intensity by 50% ... 45
Table 3. Increase Failure Intensity by 50% ... 45
Table 4. Decrease Acceleration Factors by 50% ... 47
Table 5. Increase Acceleration Factors by 50%... 47
Table 6. Decrease Effectiveness Factors by 50% ... 48
Table 7. Decrease Effectiveness Factors by 50% ... 49
Table 8. Effectiveness Factor Analysis ... 49
Table 9. Failure Modes Elimination Analysis ... 50
Table 10. Cost Analysis ... 51
Table 11. Stage 1 Indicator Matrix ... 54
Table 12. Stage 2 Indicator Matrix ... 55
Table 13. Stage 5 Indicator Matrix ... 55
Table 14．Testing Time With or Without Repair/Redesign Induced Failure Modes ... 56
Table 15. Decrease Effectiveness Factors by 50% of Modified Model ... 56
Table 16. Decrease Effectiveness Factors by 50% of Modified Model ... 57
Table 17. Protocol Research ... 69
Table 18. Full Sensitivity Experiment with Decreasing Failure Intensity By 50% ... 73
Table 19. Full Sensitivity Analysis with Decreasing Failure Intensity by 50% ... 74
Table 20. Full Sensitivity Experiment with Increasing Failure Intensity by 50% ... 75
Table 21. Full Sensitivity Analysis with Increasing Failure Intensity by 50% ... 76
Table 22. Full Sensitivity Experiment with Decreasing Acceleration Factor by 50% ... 77
Table 23. Full Sensitivity Analysis with Decreasing Acceleration Factor by 50% ... 78
Table 24. Full Sensitivity Experiment with Increasing Acceleration Factor by 50% ... 79
Table 25. Full Sensitivity Analysis with Increasing Acceleration Factor by 50% ... 80
Table 26. Full Sensitivity Experiment with Decreasing Effectiveness Factor by 50% ... 81
Table 27. Full Sensitivity Analysis with Decreasing Effectiveness Factor by 50% ... 82
Table 28. Full Sensitivity Experiment with Increasing Effectiveness Factor by 50% ... 83
Table 29. Full Sensitivity Analysis with Increasing Effectiveness Factor by 50% ... 84
Table 30. Full Sensitivity Experiment When No Failure Mode by Acceleration Factor ... 85
Table 31. Full Sensitivity Analysis When No Failure Mode by Acceleration Factor ... 86
Table 32. Full Sensitivity Experiment with Decreasing Test Cost by 50%... 87
Table 34. Full Sensitivity Experiment with Decreasing Effectiveness Factor by 50% (Redesign Induced FM Model). ... 89 Table 35. Full Sensitivity Analysis with Decreasing Effectiveness Factor by 50% (Redesign Induced FM Model) ... 90 Table 36. . Full Sensitivity Experiment with Increasing Effectiveness Factor by 50% (Redesign Induced FM Model) ... 91 Table 37. Full Sensitivity Analysis with Increasing Effectiveness Factor by 50% (Redesign Induced FM Model) ... 92
List of Figures
Figure 1. Frequency of Papers Published ... 3
Figure 2: Modeling Process ... 23
Figure 3. Testing Procedure ... 32
Figure 4. Reliability Tendency Plot ... 39
Figure 5. Modified Reliability Plot ... 58
Figure 6. Reliability Plot with All modes Failure Intensity Decreased by 50% ... 74
Figure 7. Reliability Plot with All modes Failure Intensity Increased by 50% ... 76
Figure 8. Reliability Plot with All AFs Decreased by 50% ... 78
Figure 9. Reliability Plot with All AFs Increased by 50% ... 80
Figure 10. Reliability Plot with All EFs Decreased by 50% ... 82
Figure 11. Sensitivity Plot with All EFs Decreased by 50% ... 84
Figure 12. Reliability Plot with All EFs Decreased by 50% (Redesign Induced FM Model) . 90 Figure 13. Sensitivity Plot with All EFs Increased by 50% (Redesign Induced FM Model) .. 92
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Chapter 1. Introduction
1.1. Background
Reliability engineering as a discipline became popular since just after World War II in the manufacturing field. As a critical approach in reliability improvement, the goal of accelerated tests was to improve a product’s reliability with confidence and therefore decrease maintenance costs (Klyatis & Verbitsky. 2010). While the original notion of reliability and maintainability originated primarily in the US defense sector, it was the Japanese who first utilized it extensively in the commercial sector. Today’s thriving Japanese car industry is a result of a strong focus on reliability and quality engineering. Lately, many companies have placed a renewed emphasis on reliability. Two tools, reliability growth testing, and accelerated testing have been used to help organizations measure reliability and improve their designs and products through the use of a test, analyze, and fix approach. Originally, reliability growth testing and accelerated testing have been applied separately. The possible reason for it might because “accelerated testing is usually done at the component level while reliability growth planning is more applicable at the
subsystem/system level.” (Feinberg, 1994)
Traditional reliability engineering approaches alone can no longer fully satisfy the ever increasing demands on product quality and reliability. In order to economically enhance quality and reliability, to further decrease the testing time and marginal cost, accelerated reliability growth testing (ARGT) has emerged as an area of interest. In this research, we use ARGT to help
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shorten the testing time and operational cost for a system during testing. Mathematical models on how to determine the optimal accelerated test strategy and optimize reliability for the system will be proposed in this plan.
Reliability growth is an approach that is used to eliminate failure mechanism of devices through testing (Andonova & Atanasova, 2004). Gurunatha and Siegel (2003) developed a 12-step Six-Sigma testing policy in reliability growth. Reliability growth testing has been applied in many fields, such as life time tests (Krasich, 2007, Krasich, 2011, Xing & Wu, 2011) and step stress screening (Wong, 1990, Pohl & Dietrich, 1999), to improve the effectiveness of new products. The growth of reliability comes from eliminating failure modes (FMs) through increased testing. The trade-offs between reliability and cost is determined by testing goals (Quigley & Walls, 2003). There are several famous reliability growth models that have been formulated during the past decades, for example, Duane’s model, the AMSAA (Crow) model, the IBM model, the Goel-Okumoto (G-O) model, etc. Some of these models, e.g., the G-O model, focus on software reliability growth, while others, e.g., Duane’s model and AMSAA (Crow) model, can be applied to both software and hardware systems. According to our review of the literature on accelerated reliability growth testing, while small, there has been a slight increase in the number of research papers being published in this area over the last 24 years (see below).
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Figure 1. Frequency of Papers Published
A variety of reliability growth approaches have been proposed during the past decade. For example, a common approach is to maintain a predetermined reliability level using a test, analyze and fix process, which is known as TAAF. Most of these approaches test in the nominal use environment. In this research, accelerated testing during the reliability growth process will be studied.
ARGT is designed to detect failure modes and the time to system failures in a shorter time through the use of acceleration factors (AFs) (Hu, et al., 1993, Jayatilleka & Appliances, 2006, Ye, Jiang, et al., 2013). In the literature, Yuanquan Zhou and his research team have published several ARGT papers. Their research efforts covers the introduction of ARGT (Zhou & Zhu, 2000), the definition of acceleration factors (Zhou & Zhu, 2001a), data analysis (Zhou & Zhu, 2001b, Zhou & Zhu, 2001c, Zhou & Zhu, 2001d), step stress analysis (Zhou & Zhu, 2003a), and
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Frequency Of Papers Published
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structure and performance analysis (Zhou & Zhu, 2003b). The acceleration factors can usually be divided into two main groups; environmental and operational factors (Krasich, 2004, Acevedo, 2006). The fundamental element of an ARGT approach is the requirement to determine the acceleration factors and the stress levels. Acceleration factors are defined as the influences that affect a product’s reliability most. Stress levels are determined by the ratio of normal lifetime to accelerated lifetime. The more step stress levels are implemented, the faster failures will occur. Due to the complexity of multi steps, most of the research was limited to single step stress or two step stresses (Bai & Kim, 1993, Xiong, 1999, Watkins, 2001, Alhadeed & Yang, 2002, Hassan, 2013, Kamal, Zarrin & Islam, 2013). Further, there are two common types of step stress
strategies, time step stress and failure step stress. In the literature, time step stress is more broadly used since it is easier to control.
1.2. Motivation
Our two research objectives capture our contribution to the accelerated reliability growth test literature. First, an overview will be presented on what has been done during the past 20+ years in the area of ARGT. Based on our initial examination of the literature, there has not been a systematic literature review conducted on this topic for this period. This research will fill the gap in this area. In this review, we will cover the application fields, research methods, research tendency etc., which could provide a systematic overview of this topic.
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Secondly, we will come up with a new mathematical acceleration model. Most of the research considers the situation purely at the system level (Hu, et al., 1993, Feinberg, 1994) or exclusively at the component level (Crown & Feinberg, 1998, Krasich, 2004, Zanoff &
Ekwaro-Osire, 2010). According to what we have reviewed, we noticed that there is little research that has explicitly modeled the accelerated tests on component level while measuring the reliability on system level. Since it costs less and is easier to conduct accelerated tests and reliability tests at the component level, we think it is easier to consider implementing accelerated test at the component level and nominal testing of the reliability at the system level.
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Chapter 2. Literature review
2.1. Protocol Research
In this chapter, a systematic literature review is summarized. Though ARGT does not have a long history, there are multiple models proposed in the literature that can be used to estimate reliability in this area. In addition, there are a couple of papers that focus specifically on data analysis for these types of models.
The importance of this literature review can be explained in several aspects. First, ARGT is an important part of reliability engineering yet there is little system-level literature written in this field. This literature review will help identify the void and outline opportunities to fill it. By reviewing and examining the main models and comparing their assumptions, strengths and weaknesses, we can identify areas that need further research. Second, by analyzing the existing models and applications, it can identify the possible research opportunities for future researchers. Third, since ARGT models are presented, test designers and test executers can easily find the most appropriate model to meet their demands.
Most of the ARGT models are based on existing reliability growth models. The acceleration approaches are derived from the accelerated testing literature. It is the merging of these two areas that has created the AGRT research area. In this paper, a systematic review will be presented to show the development of the ARGT field and the remaining open areas related to this topic.
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The literature review protocol used for this research was carried out in the Compendex Database, using the following protocols for key words, year, language etc. The search items used are listed below:
Criteria Protocol Description
Search Term Accelerated Reliability Growth Test Database Name: Compendex
Search field: All Fields Date: 1990 to 2013 Academic papers Exclusion
Criteria
Duplicate papers
Papers with incomplete information (titles, author, publisher, year, etc.) Papers written in language other than English
Table 1. Protocol Research Criteria
Using this protocol, 352 papers were identified that related to this topic. There are up to nine unique publications that have ARGT papers published in them. Among the papers found in the Compendex database, there are 93 of them from Institute of Electrical and Electronics
Engineering Inc. It accounts for more than half of the total number of papers. The diversity of the research field also includes but is not limited to IEEE Computer Society, Spie, and Elesevier LTD.
However, after distinguishing the pure acceleration papers, reliability papers, or specific case study papers, there were only 29 papers which were classified as ARGT papers, including ARGT applications in software systems (Okamura, Dohi, & Osaki, 2001, Wu, Zhang & Lu, 2010,
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Feng, Liu & Zheng, 2011, Wang, Wu & Li, 2011, Li, Luo & Wang, 2013). The number is relatively small for the period studied. Therefore, the average publishing rate for this topic is 0.8/year, or less than one paper per year.
There are some other papers published in languages other than English. For example, Zhou Y. and his research team published 8 ARGT papers in Chinese. Their topics covered the
theoretic foundation of ARGT, data analysis of ARGT, graphic analysis of ARGT etc. Since ARGT is an extension of Accelerated Test and Reliability Growth Test, it is not surprising to us that there are few fresh and unique authors who focused only on ARGT. After an authorship study, there were only 15 distinct new authors identified. The rest of them had published previously in either the Accelerated Test literature or the Reliability Growth Testing areas previously to publishing on ARGT. Among the researchers, Alec A. Feinberg contributed 4 papers and Milena Kasich contributed 3 papers respectively.
Within the literature, the use of acceleration factors can be divided into two areas, namely environmental and operational factors. Depending on step stress modes, Acceleration Testing is divided into failure step stress tests and time step stress tests. The acceleration factors in step stress tests could be divided into Iso (isogenous) or non-Isoapproaches. Depending on testing level, Reliability Growth Testing is usually conducted at the board level or system level, or on occasion both levels. Despite the fact that there could be a significant number of possible research combinations for ARGT, there is only a handful of ARGT topics that have been
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researched. In the next section, the existing research directions and models are examined in detail.
2.2. Existing Models
2.2.1. ISO acceleration factor
Based on MIL-KDBK-189 document, Feinberg (1994) proposed an ISO-ARGT model. The assumptions associated with this model include:
I. An effective acceleration factor, A, exists and can be estimated. II. Time is linearly compressed by the factor A.
III. Equal reliability growth is possible in an uncompressed time period t, as in the accelerated compressed time period (given as A divided by t).
( ) , (1)
( ) ( ) . (2)
is the initial MTBF, is the testing time length of stage 1, is the growth parameter, and is acceleration factor.
This is a single step acceleration model which emphasizes measuring the mean time between failures (MTBF). In the first stage, the MTBF is the initial value. In the second stage, an ISO acceleration factor is used to account for testing in an accelerated environment. Instead of
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considering all the acceleration factors separately, the effect of acceleration factors was integrated to a single one. In addition, a growth parameter was considered. The original reliability growth model is
( ) ( ) , . (3) In practice, there would be multiple acceleration factors that affect a products’ reliability. However, in this modeling approach, there is only one acceleration factor that is considered, which is A. Therefore, the model is more appropriate for a simple system or subsystem.
2.2.2. Environmental and operational acceleration factors approach
Krasich M. (2004) proposed an ARGT model that utilizes both operational and environmental stresses which is denoted as follows:
( ) ( ) ∏ ( ) ∏ . (4)
It is assumed that the lifetime of a product is T. In this model, the reliability under environmental stress is denoted as . The reliability under operational stress is denoted as
. The interaction effect of individual stresses is denoted as ( ). The subscripts and
are indices on the number of environmental acceleration factors and operation acceleration factors respectively.
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[ √ ]
. (5)
The mean demonstrated strength is while is the mean accumulated load. The corresponding variances are and respectively. According to different types of acceleration factors, the value of the acceleration factor is determined by the corresponding acceleration mode. In this model, they used thermal cycling, thermal exposure, humidity, vibration, and power cycling as possible acceleration factors. Arrhenius and power law or inverse power law models are applied in order to estimate the acceleration levels.
In the test, the stresses were applied to the system in some prescribed sequence. Although it is relatively easy to obtain the reliability under only one stress, it is often very difficult to find out the interaction index for an individual system.
Based upon this model, Krasich (2006, 2011) further discussed the data analysis and test design associated with this model. Additionally, Krasich (2014) discussed the possible errors associated with failure rate estimation due to failure modes ignorance. Specifically, she
introduces extra failure modes, design defects and random failure modes, to her model. The two new introduced failure modes are assumed to have a constant failure rate.
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Acevedo, Jackson and Kotlowitz (2006) proposed an ARGT model for electronics that focuses on critical parts. Only environmental stress factors were considered in this model. Three assumptions were made in their paper:
I. The product is repairable.
II. The product has multiple systems.
III. Repair interval is neglected in the MTBF prediction since the repair interval is assumed to be small relative to the MTBF.
They utilized a two-parameter Weibull distribution in their model. Therefore, the expected number of failures in a system is given by:
[ ( )] ∫ . (6)
Parameters estimation for and were given by
∑ ( ) , (7) ∑ ∑ ∑ ( ) . (8)
The scale and shape parameters of the Weibull distribution are and respectively. The number of failures that occur in the system is represented by . The test truncation time is and is the time of failure in system . The acceleration factor is noted as .
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Feinberg (2013) proposed a Chi-Squared accelerated reliability growth model for three test scenarios. They are a single accelerates stress test, single stress test with multiple test groups, and multiple stress test types and multiple test groups.
The corresponding failure intensity growth models are
( ( ) ) ( ( ) ) (9)
( ( ∑ ) ∑ ) ( ( ∑ ) ) (10)
∑ ∑ . (11) Modified from Duane’s growth model, Feinberg presented his growth model as
( ( ) ), (12)
. (13)
The superscript is stress type and the subscript is test number. The Chi-squared alpha value is . The number of failures for the stress test is . The fix effectiveness factor is which has a range between 0 and 1. There are units of components tested. The acceleration factor is given by for the test and is the test time.
The model was demonstrated using actual manufacturing data. Since the model concept is relatively simple, it is easy to understand and implement. Once the desired growth rate is fixed,
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the testing time can be determined from the failure intensity growth model. However, since there is only one stress considered each time, the model could be limited when multiple stresses need to be considered. In addition, the calculation process for the Chi-Square function is
computationally complex, especially when trying to automate the implementation of the model.
2.2.5. Multiple environmental stresses model
Based on the AMSAA model, Ye et al. (2013) proposed an ARGT model that includes double-stress. There are two different types of stress, temperature and non-temperature, which are considered. In this model, it assumed that the two stresses are accelerated at the same time. The other assumptions were:
I. The products experience reliability growth at normal and accelerated stress levels.
II. Stress level does not change the failure distribution but change the distribution parameters.
III. Condition of product failure mechanism does not change with stresses.
IV. Can define the relationship of reliability growth distribution and stress levels before test. V. Give time-equivalent formula under different stress levels.
Based on the Eyring Model, the model is modified to
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Reaction rate is denoted as . Absolute temperature in equation (14) is represented by T, A, B, C, D, and V are corresponding coefficients in this model. Taking the natural log of both sides, produces
( ) ( ) ( ) ( ), (15) where, , , , ( ) , ( ) .
Assuming the lifetime of the product follows an exponential distribution, and then the reliability is given by:
⁄
. (16)
Where is the normal MTBF. In this model, the value of acceleration factors was determined from the Eyring model. This model was tested for aerospace electronic products.
2.2.6. Accelerated Testing Based on the Duane Model
Wang, Zhang and Li (2013) put forward a new ARGT model based on the Duane Model. It assumes that the failures are exponentially distributed over a period of time. The cumulative number of failures is log-linear related. The traditional cumulative failure intensity for the Duane Model is given by:
. (17)
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( ) . (18)
Since and m are undetermined variables in equation (18), the estimation of and m
increases the difficulty to use the model. In this paper, in order to simplify the objective equation, the cumulative failure number is modified to
( ) . (19) The modified model only has one variable t with three parameters a, b, and c. The author suggested least square method to determine the value of a, b, and c in the content.
Therefore, the instantaneous failure rate becomes
( ) . (20)
There are three parameters, a, b and c, which are included in the model. Compared with other acceleration models, this one is relatively easy to implement. This approach is applied in the temperature acceleration case, which has demonstrated good applications. However, due to its simplicity, this model could only be applied to temperature related acceleration tests. One drawback with this model is the difficulty in obtaining valid data.
These six models presented are the main ones found in the literature and represent the current state of the art in ARGT. Although there is a large variety among reliability growth models proposed in the literature, the number of ARGT models is significantly less. There are
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two key elements in ARGT literature. The first one is to determine the acceleration stress. The second one is to determine the reliability growth policy.
In the literature, there are acceleration models for either environmental or operational factors. The relationship between acceleration factors and stress can vary considerably. It could be linear, Arrhenius, Power Law or inverse power law, Eyring, or even a mixed relationship (Krasich, 2004). The most typical one is the Arrhenius model since temperature related stresses are often the critical ones in accelerated testing. This model was used in all of the six models presented previously.
2.3. Potential Research Directions
Among the six models presented, (1), (2), (3), (5) and (6) are presented from the system level, considering acceleration testing and reliability estimation only at the system level. Model (4) considers acceleration testing and reliability evaluation both from component level and at the system level. In all of the models, the common assumption is that all the stresses work
independently. However, this might not be true in practice. In Krasich’s model, she assumed an interaction index between stresses. This makes more sense than the other models though it is difficult to determine the value of this interaction index. Based on the previous analysis, it is suggested that an approach that focuses on component acceleration testing and system reliability evaluation.
18 2.4. Source for model modification
Reviewing the literature, there are a couple of approaches to accelerated reliability growth testing. According to the literature, acceleration factors can be divided into environmental and operational factors. Krasich (2004) suggested that environmental factors include but not be limited to temperature, thermal cycles, thermal dwell, humidity, and vibration levels. Operation factors include but are not limited to power stresses, voltage variations and pressure. In this kind of accelerated test, a fixed acceleration index is attached to each factor.
Krasich’s model (2004) is presented below:
Thermal cycling acceleration:
(
) (21)
Thermal Dwell Acceleration:
[ ( )] (22) Humidity Acceleration: ( ) [ ( )] (23) Vibration Acceleration: ( ) (24)
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Another alternative approach to accelerated testing is the use of step stress testing. There are two approaches to step stress tests in the literature: time step stress and failure step stress. Time step stress is used more often in the literature than the other due to its simplicity. To simplify the analysis process, the majority of papers in the literature use only two steps. Kamal Mustafa, Shazia Zarrin and Arif-UI-Islam (2013) proposed a two-step failure time testing plan. The optimal testing time in the first stage is determined from minimizing “the asymptotic variance of the MLE of the percentile of the lifetime distribution at normal stress condition”.
Additionally, Ye et al. (2013) proposed a model, which considered both acceleration factors and step stress acceleration in their paper. They considered a two stage accelerated test with multi environmental stresses in their example. According to their results, “The accelerated reliability growth program under multiple stresses, not only can accelerate product reliability growth, effectively shorten the product development cycle, but also can get the conversion relationship between the product life and stresses, which has a wide range of applications in engineering.”
McLaren, A. E. (2011) outlines a multistage accelerated test model in her dissertation. She assumes that there are two kinds of failure modes in each stage. The weighted failure intensity of each stage is closely related to the current failure intensity and the previous failure intensity from previous stage. In her method, the known information has been used comprehensively. Our research approach builds off of this idea to estimate the failure intensity as well.
20 McLaren’s model is given as
∑ , (25)
∑ ∑ ( ) ∑ , (26) where is the failure mode, is effectiveness factor and is failure intensity.
Because
∑ ∑ , (27)
equation (27) becomes
∑ . (28) When it is assumed that ̅, which is a constant, and then substituting the observed
modes with the notation , the above equation becomes
̅ . (29) The weighted average of failure intensity of stage 2 is based upon both stage 1 and stage 2. Therefore, the length of testing time is , the weighted failure intensity is
( ) ( ) ( ) . (30) Following the same pattern,
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( ) ̅ ( ) ̅ , (31)
( ) ∑ ∑ ̅ ∑
∑ . (32)
In her research, she developed the failure intensity model iteratively. We utilize this concept to estimate failure intensity in our model.
Despite the fact that accelerated test has been researched at the component level or system level, we noticed that there is little research conducted in which it is conducted at both
component level and system level. We developed our research approach to take this into consideration in order to allow us to develop cost effective optimal test plans with multiple stages.
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Chapter 3. Proposed Modeling Framework
3.1. Problem statement
When there are new products to release, reliability testing should be conducted on them before they are placed into the market. Once the products reliability satisfies the predetermined goal with a specified level of confidence, then the tests stop. Testing costs and time are usually the two constraints which determine the number of tests and test stages. Suppose there are
stages and components in the system. Additionally, let there be a failure mode for each component in every stage. Thus, there could be times to failures occurrences. When a failure is recognized, it is not fixed until the end of the stage. Once an improvement action is implanted, it is applied the whole system. Meanwhile, engineers are working on finding the root causes to avoid similar mistakes in updated products that are produced and tested in the next testing stage. The number of failure modes and the weighting factors for each acceleration factor should be determined using the expert judgment of the design engineers or based off of
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Figure 2: Modeling Process Testing Goal Setup
AFs and FMs Defining
Constraints Identification Assumptions Modeling Model Verification Parameter Estimation Goal Satisfaction End No No Yes Yes No
24 3.2. Notation
There are a number of models proposed in the literature. Some of them are trying to maximize the reliability while the others are focusing on minimizing cost. In this paper, we emphasize on proposing an accelerated testing plan. Reliability, cost and time are considered as constraints in the model. We assume the occurrence of failures follows a Weibull distribution, which is often used to model reliability in the literature.
Currently, there is no paper in the literature that incorporate acceleration factors, failure modes, fix effectiveness factors and weighted factors all together to analyze the reliability growth process for a multi-stage test. Since the failure intensity is iterative, we obtain the failure intensity in real time, which allows timely corrective actions in the production process. We use the following notation in our model:
: Weibull scale parameter at testing stage
: Weibull shape parameter at testing stage
k: acceleration factors
m: failure modes
: design induced failure modes
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NumSys: the number of samples in testing
C: total cost
T: product release time
: fixed cost to test system level reliability at stage i
: fixed cost of per unit time for each component at stage , mode , with
acceleration factor
: acceleration factor at stage , mode , with acceleration factor
: effectiveness factor at stage , mode , with acceleration factor
:number of failures of failure mode with acceleration factor at stage
: the acceleration ratio of acceleration factor at stage i
: reliability threshold
: failure intensity in normal condition at stage , mode , with acceleration factor
: failure modes induced by redesign in normal condition at stage , mode , with acceleration factor
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: failure modes induced by redesign in acceleration condition at stage , mode , with acceleration factor
: failure intensity at stage , mode , with acceleration factor
: failure intensity at stage , mode
: failure intensity at phase i without acceleration
: failure intensity at phase i with acceleration
[ ( )]: expected individual failure intensity in time interval
: indicator variable at stage , mode , with acceleration factor
( ): reliability at time t, stage i
: testing time interval between phase i and i+1
3.3. Assumptions
To develop our model, there are 6 fundamental assumptions we make:
1) All acceleration factors work independently.
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3) For all censored faults and failures, a fix is implemented at the end of each phase. The fix is perfect and instantaneous.
4) A product’s life distribution does not change after a fix redesign, but the parameters can change.
5) The maximum number of acceleration stages is predetermined.
6) Estimates for all possible failure modes and corresponding failure intensities are available before testing.
Although the acceleration factors may influence each other in practice, in this research plan, like the majority of the papers in the literature, the assumption has been made that all the factors works independently. Due to the independence of acceleration factors, it is assumed that failure modes are independent and identically distributed as well. To simplify the problem, a TFT (test-fix-test) approach is modeled. Specifically, repair or replacement is conducted at the end of each phase instead of right after finding a failure mode. Furthermore, it is assumed that after each fix, the product’s life distribution doesn’t change. To reach the testing goal, either from the failure intensity aspect or testing cost aspect, the number of acceleration stages is then determined.
28
Before starting the test, there is a set of potential failure modes that are believed to exist in the product based on engineering analysis or from data collected from similar systems. The failures modes that have not occurred in previous data are not considered in this test planning model. Meanwhile, the failure intensity of each failure mode is evaluated according to prior statistical analysis. If the number of failures of a failure mode is greater than 1 in a testing time interval, it is assumed that the failure mode has occurred in that testing stage. Repair is
implemented at the end of each testing stage. Design improvement is devised based on the information collected from the previous stage. There is an effectiveness factor associated with each failure mode which models the reduction in the likelihood of failures of the same mode in the next testing stage. The value of effectiveness factors varies from 0 to 1. The testing time of each failure mode that did not occur in current stage is accumulated until the appearance of that mode. For example, if failure mode 1 with acceleration factor 1 does not occur in the first stage, the testing time is accumulated from stage 1 until it appears or reaches the end of the testing stage. The stopping rule of each testing stage depends on the predetermined reliability threshold. Once the reliability under the accelerated conditions reaches the threshold, testing is terminated. Based on testing goal, the number of testing levels is then determined. There is also testing costs associated with each level. Per unit of time cost varies along with the value of acceleration factors at each level.
29 3.5. Proposed Model
The objective of this testing plan is to maximize the reliability at completion of the last phase within a specified cost budget and a specified product release time. In other words, if more failure modes could be detected during testing and corrected, there will be less potential failures in the future due to preventive actions. Different from most research papers, which set a specific MTTF (mean time to fail) as a testing standard (McLinn, 1998), we focus on evaluating the system performance by using system reliability. Since the reliability is tested at the system level, the reliability function in testing stage i is written as
( ) ( ∑ ) ∑ ∑ , (33) where ( ), (34) ( ). (35)
The characteristic life of product is . The shape parameter and scale parameter are functions, denoted as h(•) and g(•), of the failure intensity. When designing a certain product, the characteristic life is usually assumed. It is assumed that is changing along with the change of failure intensity at each phase. Since the maintenance or replacement that occurs at the end of each stage is perfect, each stage could be regarded as a renewal process. Therefore, the age of the equipment is 0 at the beginning of each testing stage.
30
Before testing, the total testing cost is determined. At the component level, the testing cost is associated with the length of testing time in that stage. At the system level, there is a fixed cost which includes the cost of component replacement. Thus, the total cost for all tests should be within the cost budget.
∑ ∑ ∑ ∑ . (36) In today’s competitive global market, time to market is valued almost as much as cost in most aspects, especially in new product research. Thus, it is required that the total testing time be within a specified time limit in order to ensure the product release date is met.
∑ . (37)
When there is a single acceleration factor, it is assumed that the acceleration index at each stage is the sum of all the acceleration indices of all the factors at each stage. It is denoted as
∑ . (38)
In this test plan, multiple stages and multiple acceleration factors are considered. Following the previous pattern, the acceleration index at each stage is
∑ ∑ ( ). (39)
Our proposed model is an iterative process that takes previous stages into account in order to estimate the current acceleration index. To estimate the current acceleration intensity, an
31
effectiveness factor, which represents the likelihood that a failure model identified is eliminated in future stages, is subtracted. Since multiple stages are considered in this research, we
assume . Meanwhile, the effectiveness factor is restricted to be between 0 and 1.
Because in the assumed model, the failure intensity of the next phase is dependent on the previous stages, the model could be used as a tool in reliability prediction. Since it is assumed that testing is not ended upon component failures but reliability threshold, the testing procedures is indicated in Figure 3.
32
Figure 3. Testing Procedure 3.6. Implementation of Proposed Model
Accelerated life tests aim to find failure modes in a short time which helps predict lifetime reliability of products. In this modeling approach, we proposed an accelerated reliability growth test based on McLaren’s (2011) model and assumed an underlying Weibull distribution. In this model, we are able to predict failure intensity and reliability based on the initial failure intensity.
Failure of components
Ending of testing point?
Replacement Keep testing
Next stage testing within budget?
Replacement End testing
33
Depending on the value of the shape parameter, this model could be applied for a decreasing failure rate, increasing failure rate or constant failure rate product.
When the shape parameter , it represents a constant failure rate and the reliability of the product follows an exponential distribution. It has been shown that, for complex systems, the combination of different failure modes will often exhibit a constant failure rate during its useful life period. In the literature, in order to simplify the modeling approach process, it is common to assume the failure rate is constant.
The hazard function for the Weibull distribution, for either normal condition or an accelerated condition, is
. (40)
When , is a constant, the reliability function then becomes
( ) . (41)
When shape parameter , the product has a decreasing failure rate. Products, such as electronics often fall into this group. In order to maintain high reliability after selling products, accelerated burn-in testing is a common strategy to deal with this group of products.
When shape parameter , the product has an increasing failure rate. Products that physically wear out are included in this group.
34
In the cases when , according to probability theory,
( ) ∫ . (42)
It becomes more difficult to estimate the Weibull scale and shape parameters when the hazard function is unknown as well. Future research is needed in this area for this modeling framework.
35
Chapter 4. Multiple Cases Consideration
Depending on the value of the Weibull shape parameters, there are two different modeling approaches presented, one assumes β=1 and the other assumes β≠1. First, the exponential distribution model is presented. Then the more general case of the Weibull distribution with parameter estimation is proposed.
4.1. Exponential distribution
In this testing plan, it is assumed that acceleration starts in the first stage. The estimation of the failure intensity is an iterative process which incorporates information from current and previous stages. Suppose the initial failure intensity is , which is known before testing, for each mode without acceleration, therefore, the failure intensity under accelerated conditions in the first stage is given by:
. (43)
Since it is assumed that the times-to-failure are exponentially distributed during testing at each level, then the number of failures that occur in time interval is
. (44)
In practice, technicians know which components to fix and redesign only when failures occur. An indicator variable is assigned to each failure mode.
36
{
, ( )
is used to indicate which modes receive a fix.
Therefore the accelerated failure intensity in the first stage is
∑ ∑ . ( )
Once a failure mode occurs, which has , redesign is conducted on a system to improve the performance of the system. If failure modes do not occur in the current stage, then no effective action is implemented to improve those modes. Depending on if there is corrective action in first stage, normal condition failure intensity in the second stage is given by:
{ ( )
. (47)
Therefore, the accelerated condition failure intensity in the second stage should be
. (48)
Then the number of failures in the second stage, when a failure mode occurs in first stage, is given by:
. (49)
If not, the number of failures should be
37
Since only the failure modes that occur during testing are considered in determining testing time, the accumulated failure intensity should be
∑ ∑ . (51)
According to the indicator matrix generated in stage 2, the individual normal condition failure intensity in stage 3 is
{ ( )
. (52)
The corresponding accelerated failure intensity in the third stage is given by:
. (53)
Therefore, the corresponding number of failures should be
, ( )
where is the testing stage where failure mode with acceleration factor occurred after last testing stage.
Based on if failure modes occur or not, the cumulative failure intensity in stage 3 is given by:
38
Following this same approach, when , the individual failure intensity under normal condition at stage n is
{ ( ) ( ( ) ) ( )
( ) . (56)
Then the individual failure intensity under normal conditions should be:
. (57)
The number of failures for each failure mode is:
( ) ( ) . (58)
Where is derived as in equation (54).
If we sum over all the failure intensities in stage n, then the cumulative failure intensity should be:
∑ ∑ . (59)
It is not difficult to prove that when accelerating for more time, the final failure intensity is lower. In other words, products have better performance when they go through accelerated testing and specific failure modes are mitigated. In this model, which simulates the behavior of the product in normal and accelerated environments, we assume products are tested at 5 different levels. The red curves in Figure 4 represent the reliability under normal conditions after each fix and redesign. The blue curves characterize reliability in the accelerated condition of the product
39
during that specific stage. Red curves are normal condition reliability and blue curves represent acceleration condition reliability for all following plots. When the reliability in the testing stage reaches a threshold value of (0.6), testing in the current stage is terminated. Due to the effect of product redesign, the reliability under normal conditions becomes flatter. Therefore, the
reliability of product has been improved. Correspondingly, the testing time under accelerated conditions has been extended because of the increasing of the acceleration factors
Figure 4. Reliability Tendency Plot 4.2. Weibull Distribution
In the previous two cases, we assumed that the Weibull shape parameter, , under this condition, the reliability function is exponentially distributed with a constant hazard function at each stress level. However, when , the hazard function is changes with time. Assume the
40
reliability function follows a Weibull distribution over all the stages. In each stage, the shape parameter is fixed while the scale parameter varies over time. Therefore, in the first stage, the reliability is
( ) ( ) . (60)
The reliability function in the second stage is given by:
( ) ( ) . ( )
Unlike the exponential distribution, where the hazard function is a constant, the Weibull distribution has hazard function that varies with time, which is given as:
( ) . (62)
Assume we have some understanding of the distribution of failure modes before testing, denoted as ( ), the corresponding failure intensity distribution under accelerated conditions should be
( ) ( ) . (63)
Therefore, the expected failure intensity for a specific time interval is:
[ ( )] ∫ ( ) . (64) Follow the same approach used earlier for the Exponential distribution, if a failure mode occurs, it is included in estimating the testing time. Also, fix or redesign is implemented to
41
improve performance of products. If not, no maintenance or fix performs. The number of failures that occur for each failure mode is denoted as . Therefore,
[ ( )] . (65)
As before an indicator
{
, (66)
is associated with each failure mode, which has. Therefore, the failure intensity in level one should be
( ) ∑ ∑ ( ) . (67)
Combing (62) and (67) yields the expression
∑ ∑
( )
. (68)
Integrating both sides with small , , then we can estimate the value of and . Since it is assumed that does not change over time, only should update to in the second stage.
In the second stage, which depends on the value of the indicator function, the normal condition failure intensity distribution is similar to equation (47). Once a failure mode occurs, then , when the reliability reaches the threshold, perfect replacement is carried out on the associated component. If failure modes do not occur in the current stage, then no effective action
42
is implemented to improve those modes. Depending on if there is corrective action in first stage, normal condition failure intensity in the second stage is given by:
( ) { ( )( )
( ) . (69)
Therefore, the accelerated condition failure intensity is:
( ) ( ) . (70)
To calculate the number of failures in the second stage, we need to find the expected value of the failure intensity function. Then, we have
[ ( )] ∫ ( ) . (71) Thus, the number of failures of each failure mode is
[ ( )] . (72)
Depending on the number of failures which occur during testing, the indicators are
{
. (73)
Similarly, the failure intensity in stage 2 is expressed as
( ) ∑ ∑ ( ) . (74) Also, combining equation (62) and equation (74), then we have
43
∑ ∑
( )
, (75)
which could be used to update the scale parameter in stage 2.
Following this same pattern until stage , then we can update as
∑ ∑
( )
. (76)
Then the corresponding reliability function is
( ) ( ∑ ) . (77)
44
Chapter 5. Modeling Analysis and Sensitivity Analysis
In this chapter, we simulate the behavior of our ARGT model. We use our modeling framework to identify the most significant failure modes with acceleration factors, and then assess the effect of the various model parameters on the system reliability behavior by using sensitivity. The parameters investigated include: acceleration factors, effectiveness factors, potential failure modes, and initial failure intensity.
5.1. Failure intensity analysis
In Table 2, sensitivity analysis associated with failure intensity is presented. The analysis is based on decreasing the baseline failure intensity of each failure mode by 50%. For example, if the baseline failure intensity in the first stage was decreased by 50% for failure mode 1, the overall failure intensity under normal condition (λ_Ratio) decreases by -0.94%. The difference of failure intensity under acceleration and normal condition are denoted as Δλ(A) and Δλ(N)
respectively. Corresponding to this test, the testing time has increased by approximately 47 units of testing time and cost has increased by 500. Also, it is seen that when the ratio of failure intensity is positive, which means the final failure intensity increases, the cumulative time and cost are decreased. Therefore, it is not difficult to notice that decreasing the failure intensity results in increasing the accumulative testing time (ΔAccT) and testing cost (ΔAccC). Correspondingly, testing time ratio (T_Ratio) and testing financial cost ratio (C_Ratio) are increased.When decreasing failure modes 6 and 7, the final failure intensity has decreased the
45
most. Comparing the results from failure modes 6 (FM6) and 2 (FM2), when decreasing failure mode 2 by 50%, it costs more time and money to complete 5 acceleration stages then decreasing failure mode 6. Also, the final failure intensity is higher than the case when decreasing failure mode 6. Then in this case, decreasing failure mode 6 is profitable then decreasing failure mode 2.
ΔAccT ΔAccC T_Ratio C_Ratio Δλ(A) Δλ(N) λ_Ratio FM1 38.40 410 6.83% 5.08% -1.68E-04 -1.21E-06 -2.35% FM2 50.71 558 9.02% 6.91% -2.64E-04 -1.27E-06 -2.47% FM3 22.50 224 4.00% 2.77% -1.70E-05 -1.68E-06 -3.27% FM4 32.82 352 5.84% 4.36% -1.49E-04 -1.58E-06 -3.06% FM5 32.23 329 5.73% 4.07% -1.09E-04 -1.18E-06 -2.29% FM6 30.04 312 5.34% 3.86% -1.48E-04 -3.11E-06 -6.03% FM7 43.68 485 7.77% 6.00% -2.66E-04 -3.42E-06 -6.64% FM8 27.54 267 4.90% 3.30% -6.24E-05 -2.35E-07 -0.46%
Table 2. Decrease Failure Intensity by 50%
Similar results are obtained from Table 3, which shows the impact of increasing of each mode’s intensity by 50%. When increasing the failure intensity, the testing time and testing cost are decreased greatly if the failure modes are significant in the testing period. Examining the ratios, failure modes 4 and 6 affect the failure intensity most. Failure mode 8 decreases the testing time as much as failure mode 6, while the failure intensity is not increased as much as failure mode 6.
ΔAccT ΔAccC T_Ratio C_Ratio Δλ(A) Δλ(N) λ_Ratio FM1 -27.13 -269 -4.82% -3.33% -1.83E-05 1.90E-06 3.69% FM2 -32.13 -331 -5.71% -4.10% 5.24E-07 3.23E-07 0.63% FM3 -29.68 -322 -5.28% -3.98% 4.49E-05 2.99E-06 5.80% FM4 -42.12 -478 -7.49% -5.92% 2.01E-04 4.86E-06 9.43%
46
ΔAccT ΔAccC T_Ratio C_Ratio Δλ(A) Δλ(N) λ_Ratio FM5 -42.51 -467 -7.56% -5.78% 1.49E-04 2.98E-06 5.78% FM6 -48.81 -568 -8.68% -7.03% 3.39E-04 5.25E-06 10.19% FM7 -38.68 -440 -6.88% -5.45% 1.81E-04 3.72E-06 7.22% FM8 -45.67 -516 -8.12% -6.39% 2.09E-04 3.11E-06 6.03%
Table 4. Increase Failure Intensity by 50% (Cont.) 5.2. Acceleration factors analysis
In Table 4 and Table 5, sensitivity analysis of changes in the acceleration factors (AF) is presented. The value of acceleration factors 1(AF1), 2 (AF2), 3 (AF3) and 4 (AF4) are derived from inverse power law model but with different index, which will be introduced in Chapter 7. Aacceleration factors 5 (AF5) and 6 (AF6) are derived from Arrhenius relationship, as shown in equation (89). Acceleration factors 7 (AF7) and 8 (AF8) are derived from equation (91). Specific introduction of the three-parameter estimation equation will be introduced in Chapter 7. When decreasing acceleration factor 1, which is the AF associated with failure mode 1 and is modeled by an acceleration model, one finds it affects the final failure intensity the most. This is due to the reduction of testing time that is achieved due to this high acceleration rate. When increasing acceleration factors by 50%, acceleration factor 2 (AF2) affects the failure intensity most. As we can see, though failure intensity and testing time decrease in this case, the testing cost increases. Since it is assumed that testing cost of component level is associated with acceleration factors, increasing acceleration factors results in an increase of testing cost. The cost is even higher than the cost saving from the shorter testing time. Depending on financial and product release time constraints, a testing strategy could be determined based on the following two tables.
47
ΔAccT ΔAccC T_Ratio C_Ratio Δλ(A) Δλ(N) λ_Ratio AF1 24.55 -287 4.36% -3.55% 2.40E-05 2.36E-05 46.27% AF2 42.25 -224 7.51% -2.77% -2.74E-04 -4.15E-06 -8.15% AF3 28.53 -525 5.07% -6.50% -7.54E-05 9.47E-06 18.59% AF4 41.42 -526 7.37% -6.51% -2.48E-04 2.41E-07 0.47% AF5 37.64 316 6.69% 3.91% -4.38E-05 -2.63E-06 -5.16% AF6 40.41 373 7.19% 4.62% -1.57E-04 2.66E-06 5.22% AF7 32.22 307 5.73% 3.80% -1.79E-04 2.69E-07 0.53% AF8 26.89 197 4.78% 2.44% -2.12E-05 4.58E-07 0.90%
Table 5. Decrease Acceleration Factors by 50%
ΔAccT ΔAccC T_Ratio C_Ratio Δλ(A) Δλ(N) λ_Ratio AF1 -42.09 34 -7.48% 0.42% 2.11E-04 1.97E-06 3.83% AF2 -40.40 134 -7.18% 1.66% 2.75E-04 -5.08E-07 -0.99% AF3 -22.53 524 -4.01% 6.49% 2.76E-05 -1.64E-06 -3.18% AF4 -46.12 313 -8.20% 3.88% 2.07E-04 -2.38E-07 -0.46% AF5 -50.17 -509 -8.92% -6.30% 1.39E-04 -9.36E-07 -1.82% AF6 -41.49 -410 -7.38% -5.08% 1.11E-04 -1.17E-06 -2.27% AF7 -31.58 -305 -5.61% -3.78% 5.98E-05 -5.90E-08 -0.11% AF8 -34.15 -309 -6.07% -3.83% -2.12E-05 -1.23E-06 -2.38%
Table 6. Increase Acceleration Factors by 50%
Also we could see in Table 4 that when decreasing the acceleration factor 2 and 5, the last stage failure intensity is decreasing. This is because the due to the extension of testing time, more failure modes could occur and associated failure modes are well fixed. Therefore, there is lower failure intensity after that stage.
48 5.3. Effectiveness factors analysis
Table 7 and Table 8 examine the sensitivity of the model to changes in the fix effectiveness factor (EF). It is not difficult to see that the EFs affect the failure intensity the most when compared to all of the other model parameters. The failure intensity increased as much as 74% when effectiveness factor of failure mode 1(FM1) was decreased by 50%. Correspondingly, testing time and cost are reduced significantly. Similarly, increasing the effectiveness factor of failure mode 2 by 50%, decreases the failure intensity by as much as 22%. Therefore, improving the value of effectiveness factor should be explored as part of the reliability growth process as it has the most significant impact on cost, time and the final system reliability of the products that will be delivered to the field. In practice, this requires that the testers correctly find the root causes of the failures and make design changes that eliminate or mitigate the occurrence of the failure mode.
ΔAccT ΔAccC T_Ratio C_Ratio Δλ(A) Δλ(N) λ_Ratio FM1 -147.92 -1851 -24.62% -21.61% 0.0016 3.80E-05 74.67% FM2 -138.10 -1711 -22.98% -19.98% 0.0013 2.46E-05 48.37% FM3 -126.89 -1578 -21.12% -18.43% 0.0012 2.45E-05 48.05% FM4 -139.82 -1719 -23.27% -20.07% 0.0012 1.12E-05 21.98% FM5 -92.30 -1103 -15.36% -12.88% 0.0005 5.31E-06 10.43% FM6 -100.66 -1209 -16.75% -14.12% 0.0005 4.30E-06 8.44% FM7 -52.28 -629 -8.70% -7.34% 0.0002 -9.26E-08 -0.18% FM8 -95.00 -1171 -15.81% -13.68% 0.0006 1.47E-08 0.03%
49
ΔAccT ΔAccC T_Ratio C_Ratio Δλ(A) Δλ(N) λ_Ratio FM1 28.19 310 4.69% 3.62% 0.0001 -1.09E-05 -21.41% FM2 66.64 814 11.09% 9.50% -0.0003 -1.13E-05 -22.13% FM3 22.27 244 3.71% 2.85% 0.0000 -3.85E-06 -7.56% FM4 27.49 299 4.58% 3.49% 0.0000 -2.95E-06 -5.79% FM5 97.78 1179 16.27% 13.77% -0.0004 -3.35E-06 -6.58% FM6 75.59 909 12.58% 10.62% -0.0003 -2.12E-06 -4.17% FM7 41.56 506 6.92% 5.91% -0.0002 -1.37E-07 -0.27% FM8 38.09 444 6.34% 5.18% -0.0001 -1.06E-07 -0.21%
Table 8. Decrease Effectiveness Factors by 50%
Table 8 illustrates the effect of the effectiveness function on the modeling framework when EF equals to 0 (EF-0), 0.5 (EF-0.5), 0.9 (EF-0.9), and 1(EF-1). It is clear that the larger EF the lower the corresponding failure intensity is in subsequent stages. However, the testing time and cost are much higher to maintain a high level of EF. Therefore, we should choose appropriate EFs according to some specified testing budget in practice.
AccT AccC T_Ratio C_Ratio λ(A) λ(N)
Baseline 600.871 8562 0.0018 5.09E-05 0.0018 5.09E-05 EF-0 149.866 3244 0.0294 4.45E-04 0.0294 4.45E-04 EF-0.5 271.061 5617 0.0057 1.09E-04 0.0057 1.09E-04 EF-0.9 1409.336 17891 0.0008 1.77E-05 0.0008 1.77E-05 EF-1 9614.779 120444 0.0001 4.52E-07 0.0001 4.52E-07
Table 9. Effectiveness Factor Analysis 5.4. Failure modes elimination analysis
Table 10 illustrates the sensitivity analysis associated with eliminating failure modes due to increased levels of accelerated testing. If failures associated with a specific AF are eliminated, the most significant acceleration factor produces the largest decrease of failure intensity. When eliminating failure modes associated with AF2, the subsequent failure intensity decreases the
50
most while the testing cost increases. Similarly, when eliminating failure modes associated with AF1, the failure intensity decreases up by 28% while testing cost increases by less than half of the increase for AF2. There are two interesting cases that warrant further discussion. For AF7 and AF8, in this experiment, no failures occur during testing and the resulting failure intensities go up after the testing. This is because the increased accelerated of testing time increases the likelihood of failure modes in subsequent stages of testing. Based on this information, careful selection of an improvement plan should be made to reach the reliability growth goal.
ΔAccT ΔAccC T_Ratio C_Ratio Δλ(A) Δλ(N) λ_Ratio AF1 65.53 722 10.91% 8.43% -0.0001 -1.44E-05 -28.22% AF2 136.21 1581 22.67% 18.47% -0.0004 -1.72E-05 -33.67% AF3 57.05 630 9.49% 7.36% -0.0001 -5.03E-06 -9.88% AF4 75.25 786 12.52% 9.18% -0.0001 -5.04E-06 -9.89% AF5 214.01 2391 35.62% 27.93% -0.0006 -7.17E-06 -14.07% AF6 135.63 1488 22.57% 17.38% -0.0004 -2.90E-06 -5.70% AF7 82.75 935 13.77% 10.92% -0.0003 5.42E-07 1.06% AF8 68.88 705 11.46% 8.24% -0.0001 5.66E-07 1.11%
Table 10. Failure Modes Elimination Analysis 5.5. Cost analysis
Testing cost is one of the main constraints in our model. Accessing the cost sensitivity of each failure mode is meaningful when a decision is needed to improve the economics associated with the test plan. In our testing model, the testing cost is influenced by the level of acceleration as well as total testing time. Since the failure intensity of each failure mode doesn’t change, the testing time remains the same in each testing level. Therefore, testing cost is only associated with
51
acceleration factors at this point. Table 11 presents the cost analysis associated with each acceleration factor.
AF1 AF2 AF3 AF4 AF5 AF6 AF7 AF8
ΔAccC -552 -709 -858 -980 -34 -39 -53 -57
C_Ratio -6.44% -8.28% -10.02% -11.44% -0.40% -0.46% -0.62% -0.67% Table 11. Cost Analysis
In Table 10, we can see that the cost of AF4 affects testing cost the most while AF 5, 6, 7, and 8 has very little influence. Therefore, in real testing, we should try to minimize the effect of acceleration factors which are most sensitive to tests. Instead of saving cost, testing cost goes up by the same amount when increasing testing cost of each acceleration factor by 50%.
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Chapter 6. Design Induced Failure Modes Model
6.1. Modified model
In the previous proposed model, it is assumed that all possible failure modes could be
predicted before testing. Therefore, no new failure modes are induced by redesigning the product after failures have occurred. However, in reality, design/maintenance induced failure modes are very common. Building on the model that was discussed in Chapter 4, design induced failure modes were incorporated into the model presented below.
The first stage of the revised model is the same since there is no design modification. Therefore, the testing time at the first level is determined based on the failure modes that occurred. After testing phase 1, design improvement is made according to the failures that appeared in stage 1. Different from the previous model, design induced failure modes are introduced after each redesign change at stage . The failure intensity of a redesign induced failure mode is . Since most of the time designers can predict new failure modes based upon design changes, effectiveness actions are taken before starting a new testing level. Therefore, the failure intensity of design induced failure modes are:
( ( ) ) . (78)
The corresponding number of failures is given by:
